Textbook Question
Solve each equation in Exercises 47–64 by completing the square.
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1. Equations & Inequalities
Choosing a Method to Solve Quadratics
2:07 minutesProblem 85
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice.
Verified step by step guidance1
Rewrite the given equation to standard quadratic form by moving all terms to one side: \$5x^2 + 2 = 11x\( becomes \)5x^2 - 11x + 2 = 0$.
Identify the coefficients in the quadratic equation \(ax^2 + bx + c = 0\): here, \(a = 5\), \(b = -11\), and \(c = 2\).
Use the quadratic formula to solve for \(x\): \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots.
Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula and simplify under the square root and the entire expression to find the solutions for \(x\).
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equations
A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0. Solving such equations involves finding the values of x that satisfy the equation. Recognizing the standard form is essential for applying appropriate solution methods.
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Rearranging Equations
Rearranging involves moving all terms to one side to set the equation equal to zero. This step is crucial for quadratic equations because it allows the equation to be expressed in standard form, enabling the use of factoring, completing the square, or the quadratic formula.
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Methods for Solving Quadratic Equations
Common methods include factoring, completing the square, and using the quadratic formula. Factoring works when the quadratic can be expressed as a product of binomials. The quadratic formula applies universally and is derived from completing the square.
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